It's very easy to remember.
Well, we will not go far, we will immediately consider the inverse function. What is the inverse of the exponential function? Logarithm:
In our case, the base is a number:
Such a logarithm (that is, a logarithm with a base) is called a “natural” one, and we use a special notation for it: we write instead.
What is equal to? Of course, .
The derivative of the natural logarithm is also very simple:
Examples:
- Find the derivative of the function.
- What is the derivative of the function?
Answers: The exponent and the natural logarithm are functions that are uniquely simple in terms of the derivative. Exponential and logarithmic functions with any other base will have a different derivative, which we will analyze later, after we go through the rules of differentiation.
Differentiation rules
What rules? Another new term, again?!...
Differentiation is the process of finding the derivative.
Only and everything. What is another word for this process? Not proizvodnovanie... The differential of mathematics is called the very increment of the function at. This term comes from the Latin differentia - difference. Here.
When deriving all these rules, we will use two functions, for example, and. We will also need formulas for their increments:
There are 5 rules in total.
The constant is taken out of the sign of the derivative.
If - some constant number (constant), then.
Obviously, this rule also works for the difference: .
Let's prove it. Let, or easier.
Examples.
Find derivatives of functions:
- at the point;
- at the point;
- at the point;
- at the point.
Solutions:
- (the derivative is the same at all points, since it's a linear function, remember?);
Derivative of a product
Everything is similar here: we introduce a new function and find its increment:
Derivative:
Examples:
- Find derivatives of functions and;
- Find the derivative of a function at a point.
Solutions:
Derivative of exponential function
Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just the exponent (have you forgotten what it is yet?).
So where is some number.
We already know the derivative of the function, so let's try to bring our function to a new base:
To do this, we use a simple rule: . Then:
Well, it worked. Now try to find the derivative, and don't forget that this function is complex.
Happened?
Here, check yourself:
The formula turned out to be very similar to the derivative of the exponent: as it was, it remains, only a factor appeared, which is just a number, but not a variable.
Examples:
Find derivatives of functions:
Answers:
This is just a number that cannot be calculated without a calculator, that is, it cannot be written in a simpler form. Therefore, in the answer it is left in this form.
Note that here is the quotient of two functions, so we apply the appropriate differentiation rule:
In this example, the product of two functions:
Derivative of a logarithmic function
Here it is similar: you already know the derivative of the natural logarithm:
Therefore, to find an arbitrary from the logarithm with a different base, for example, :
We need to bring this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:
Only now instead of we will write:
The denominator turned out to be just a constant (a constant number, without a variable). The derivative is very simple:
Derivatives of the exponential and logarithmic functions are almost never found in the exam, but it will not be superfluous to know them.
Derivative of a complex function.
What is a "complex function"? No, this is not a logarithm, and not an arc tangent. These functions can be difficult to understand (although if the logarithm seems difficult to you, read the topic "Logarithms" and everything will work out), but in terms of mathematics, the word "complex" does not mean "difficult".
Imagine a small conveyor: two people are sitting and doing some actions with some objects. For example, the first wraps a chocolate bar in a wrapper, and the second ties it with a ribbon. It turns out such a composite object: a chocolate bar wrapped and tied with a ribbon. To eat a chocolate bar, you need to do the opposite steps in reverse order.
Let's create a similar mathematical pipeline: first we will find the cosine of a number, and then we will square the resulting number. So, they give us a number (chocolate), I find its cosine (wrapper), and then you square what I got (tie it with a ribbon). What happened? Function. This is an example of a complex function: when, in order to find its value, we do the first action directly with the variable, and then another second action with what happened as a result of the first.
In other words, A complex function is a function whose argument is another function: .
For our example, .
We may well do the same actions in reverse order: first you square, and then I look for the cosine of the resulting number:. It is easy to guess that the result will almost always be different. An important feature of complex functions: when the order of actions changes, the function changes.
Second example: (same). .
The last action we do will be called "external" function, and the action performed first - respectively "internal" function(these are informal names, I use them only to explain the material in simple language).
Try to determine for yourself which function is external and which is internal:
Answers: The separation of inner and outer functions is very similar to changing variables: for example, in the function
- What action will we take first? First we calculate the sine, and only then we raise it to a cube. So it's an internal function, not an external one.
And the original function is their composition: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: .
we change variables and get a function.
Well, now we will extract our chocolate - look for the derivative. The procedure is always reversed: first we look for the derivative of the outer function, then we multiply the result by the derivative of the inner function. For the original example, it looks like this:
Another example:
So, let's finally formulate the official rule:
Algorithm for finding the derivative of a complex function:
It seems to be simple, right?
Let's check with examples:
Solutions:
1) Internal: ;
External: ;
2) Internal: ;
(just don’t try to reduce by now! Nothing is taken out from under the cosine, remember?)
3) Internal: ;
External: ;
It is immediately clear that there is a three-level complex function here: after all, this is already a complex function in itself, and we still extract the root from it, that is, we perform the third action (put chocolate in a wrapper and with a ribbon in a briefcase). But there is no reason to be afraid: anyway, we will “unpack” this function in the same order as usual: from the end.
That is, first we differentiate the root, then the cosine, and only then the expression in brackets. And then we multiply it all.
In such cases, it is convenient to number the actions. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's look at an example:
The later the action is performed, the more "external" the corresponding function will be. The sequence of actions - as before:
Here the nesting is generally 4-level. Let's determine the course of action.
1. Radical expression. .
2. Root. .
3. Sinus. .
4. Square. .
5. Putting it all together:
DERIVATIVE. BRIEFLY ABOUT THE MAIN
Function derivative- the ratio of the increment of the function to the increment of the argument with an infinitesimal increment of the argument:
Basic derivatives:
Differentiation rules:
The constant is taken out of the sign of the derivative:
Derivative of sum:
Derivative product:
Derivative of the quotient:
Derivative of a complex function:
Algorithm for finding the derivative of a complex function:
- We define the "internal" function, find its derivative.
- We define the "external" function, find its derivative.
- We multiply the results of the first and second points.
Do you think there is still a lot of time before the exam? Is it a month? Two? Year? Practice shows that the student copes best with the exam if he began to prepare for it in advance. There are a lot of difficult tasks in the Unified State Examination that stand in the way of a student and a future applicant to the highest scores. These obstacles need to be learned to overcome, besides, it is not difficult to do this. You need to understand the principle of working with various tasks from tickets. Then there will be no problems with the new ones.
Logarithms at first glance seem incredibly complex, but upon closer analysis, the situation becomes much simpler. If you want to pass the exam with the highest score, you should understand the concept in question, which we propose to do in this article.
First, let's separate these definitions. What is a logarithm (log)? This is an indicator of the power to which the base must be raised in order to obtain the specified number. If it is not clear, we will analyze an elementary example.
In this case, the base below must be raised to the second power to get the number 4.
Now let's deal with the second concept. The derivative of a function in any form is called a concept that characterizes the change in a function at a given point. However, this is a school curriculum, and if you experience problems with these concepts separately, it is worth repeating the topic.
Derivative of the logarithm
In the USE assignments on this topic, several tasks can be cited as an example. Let's start with the simplest logarithmic derivative. We need to find the derivative of the following function.
We need to find the next derivative
There is a special formula.
In this case x=u, log3x=v. Substitute the values from our function into the formula.
The derivative of x will be equal to one. The logarithm is a little more difficult. But you will understand the principle if you just substitute the values. Recall that the derivative of lg x is the derivative of the decimal logarithm, and the derivative of ln x is the derivative of the natural logarithm (based on e).
Now just substitute the obtained values into the formula. Try it yourself, then check the answer.
What could be the problem here for some? We have introduced the concept of the natural logarithm. Let's talk about it, and at the same time figure out how to solve problems with it. You will not see anything complicated, especially when you understand the principle of its operation. You should get used to it, as it is often used in mathematics (especially in higher educational institutions).
Derivative of the natural logarithm
At its core, this is the derivative of the logarithm to the base e (this is an irrational number that equals approximately 2.7). In fact, ln is very simple, which is why it is often used in mathematics in general. Actually, solving the problem with him will not be a problem either. It is worth remembering that the derivative of the natural logarithm to the base e will be equal to one divided by x. The solution of the following example will be the most indicative.
Imagine it as a complex function consisting of two simple ones.
enough to transform
We are looking for the derivative of u with respect to x
Let's continue with the second
We use the method of solving the derivative of a complex function by substituting u=nx.
What happened in the end?
Now let's remember what n meant in this example? This is any number that can occur in the natural logarithm before x. It is important for you to understand that the answer does not depend on it. Substitute anything, the answer will still be 1/x.
As you can see, there is nothing complicated here, it is enough just to understand the principle in order to quickly and efficiently solve problems on this topic. Now you know the theory, it remains to consolidate in practice. Practice solving problems to remember the principle of solving them for a long time. You may not need this knowledge after graduation, but on the exam it will be more relevant than ever. Good luck to you!
Proof and derivation of formulas for the derivative of the natural logarithm and the logarithm in base a. Examples of calculating derivatives of ln 2x, ln 3x and ln nx. Proof of the formula for the derivative of the nth order logarithm by the method of mathematical induction.
ContentSee also: Logarithm - properties, formulas, graph
Natural logarithm - properties, formulas, graph
Derivation of formulas for derivatives of the natural logarithm and the logarithm in base a
The derivative of the natural logarithm of x is equal to one divided by x:
(1)
(lnx)′ =.
The derivative of the logarithm to the base a is equal to one divided by the variable x multiplied by the natural logarithm of a :
(2)
(log x)′ =.
Proof
Let there be some positive number not equal to one. Consider a function that depends on the variable x , which is a base logarithm:
.
This function is defined with . Let's find its derivative with respect to x . By definition, the derivative is the following limit:
(3)
.
Let's transform this expression to reduce it to known mathematical properties and rules. To do this, we need to know the following facts:
A) Properties of the logarithm. We need the following formulas:
(4)
;
(5)
;
(6)
;
B) Continuity of the logarithm and property of limits for a continuous function:
(7)
.
Here, is some function that has a limit and this limit is positive.
IN) The meaning of the second wonderful limit:
(8)
.
We apply these facts to our limit. First we transform the algebraic expression
.
To do this, we apply properties (4) and (5).
.
We use property (7) and the second remarkable limit (8):
.
And finally, apply property (6):
.
base logarithm e called natural logarithm. It is marked like this:
.
Then ;
.
Thus, we have obtained formula (2) for the derivative of the logarithm.
Derivative of the natural logarithm
Once again, we write out the formula for the derivative of the logarithm in base a:
.
This formula has the simplest form for the natural logarithm, for which , . Then
(1)
.
Because of this simplicity, the natural logarithm is very widely used in calculus and other areas of mathematics related to differential calculus. Logarithmic functions with other bases can be expressed in terms of the natural logarithm using property (6):
.
The base derivative of the logarithm can be found from formula (1) if the constant is taken out of the differentiation sign:
.
Other ways to prove the derivative of the logarithm
Here we assume that we know the formula for the derivative of the exponent:
(9)
.
Then we can derive the formula for the derivative of the natural logarithm, given that the logarithm is the inverse of the exponent.
Let us prove the formula for the derivative of the natural logarithm, applying the formula for the derivative of the inverse function:
.
In our case . The inverse of the natural logarithm is the exponent:
.
Its derivative is determined by formula (9). Variables can be denoted by any letter. In formula (9), we replace the variable x with y:
.
Since , then
.
Then
.
The formula has been proven.
Now we prove the formula for the derivative of the natural logarithm using rules for differentiating a complex function. Since the functions and are inverse to each other, then
.
Differentiate this equation with respect to the variable x :
(10)
.
The derivative of x is equal to one:
.
We apply the rule of differentiation of a complex function:
.
Here . Substitute into (10):
.
From here
.
Example
Find derivatives of ln 2x, ln 3x And ln nx.
The original functions have a similar form. Therefore, we will find the derivative of the function y = log nx. Then we substitute n = 2 and n = 3 . And, thus, we obtain formulas for derivatives of ln 2x And ln 3x .
So, we are looking for the derivative of the function
y = log nx
.
Let's represent this function as a complex function consisting of two functions:
1)
Variable dependent functions : ;
2)
Variable dependent functions : .
Then the original function is composed of the functions and :
.
Let's find the derivative of the function with respect to the variable x:
.
Let's find the derivative of the function with respect to the variable:
.
We apply the formula for the derivative of a complex function.
.
Here we have substituted .
So we found:
(11)
.
We see that the derivative does not depend on n. This result is quite natural if we transform the original function using the formula of the logarithm of the product:
.
- is a constant. Its derivative is zero. Then, according to the rule of differentiation of the sum, we have:
.
; ; .
Derivative of logarithm modulo x
Let's find the derivative of another very important function - the natural logarithm of the x module:
(12)
.
Let's consider the case. Then the function looks like:
.
Its derivative is determined by formula (1):
.
Now consider the case . Then the function looks like:
,
Where .
But we also found the derivative of this function in the above example. It does not depend on n and is equal to
.
Then
.
We combine these two cases into one formula:
.
Accordingly, for the logarithm to the base a, we have:
.
Higher order derivatives of the natural logarithm
Consider the function
.
We found its first order derivative:
(13)
.
Let's find the second order derivative:
.
Let's find the derivative of the third order:
.
Let's find the derivative of the fourth order:
.
It can be seen that the nth order derivative has the form:
(14)
.
Let us prove this by mathematical induction.
Proof
Let us substitute the value n = 1 into formula (14):
.
Since , then for n = 1
, formula (14) is valid.
Let us assume that formula (14) is satisfied for n = k . Let us prove that it follows from this that the formula is valid for n = k + 1 .
Indeed, for n = k we have:
.
Differentiate with respect to x :
.
So we got:
.
This formula coincides with formula (14) for n = k + 1
. Thus, from the assumption that formula (14) is valid for n = k, it follows that formula (14) is valid for n = k + 1
.
Therefore, formula (14), for the nth order derivative, is valid for any n .
Higher-order derivatives of the logarithm to base a
To find the nth derivative of the base logarithm a , you need to express it in terms of the natural logarithm:
.
Applying formula (14), we find the nth derivative:
.
complex derivatives. Logarithmic derivative.
Derivative of exponential function
We continue to improve our differentiation technique. In this lesson, we will consolidate the material covered, consider more complex derivatives, and also get acquainted with new tricks and tricks for finding the derivative, in particular, with the logarithmic derivative.
Those readers who have a low level of preparation should refer to the article How to find the derivative? Solution examples which will allow you to raise your skills almost from scratch. Next, you need to carefully study the page Derivative of a complex function, understand and resolve All the examples I have given. This lesson is logically the third in a row, and after mastering it, you will confidently differentiate fairly complex functions. It is undesirable to stick to the position “Where else? Yes, and that's enough! ”, Since all the examples and solutions are taken from real tests and are often found in practice.
Let's start with repetition. At the lesson Derivative of a complex function we have considered a number of examples with detailed comments. In the course of studying differential calculus and other sections of mathematical analysis, you will have to differentiate very often, and it is not always convenient (and not always necessary) to paint examples in great detail. Therefore, we will practice in the oral finding of derivatives. The most suitable "candidates" for this are derivatives of the simplest of complex functions, for example:
According to the rule of differentiation of a complex function 
:
When studying other matan topics in the future, such a detailed record is most often not required, it is assumed that the student is able to find similar derivatives on autopilot. Let's imagine that at 3 o'clock in the morning the phone rang, and a pleasant voice asked: "What is the derivative of the tangent of two x?". This should be followed by an almost instantaneous and polite response: 
.
The first example will be immediately intended for an independent solution.
Example 1
Find the following derivatives orally, in one step, for example: . To complete the task, you only need to use table of derivatives of elementary functions(if she hasn't already remembered). If you have any difficulties, I recommend re-reading the lesson Derivative of a complex function.
, , ,
, 
, , 
, , ,
, , 
,
, , 
,
, , ,
, 
,
Answers at the end of the lesson
Complex derivatives
After preliminary artillery preparation, examples with 3-4-5 attachments of functions will be less scary. Perhaps the following two examples will seem complicated to some, but if they are understood (someone suffers), then almost everything else in differential calculus will seem like a child's joke.
Example 2
Find the derivative of a function 
As already noted, when finding the derivative of a complex function, first of all, it is necessary Right UNDERSTAND INVESTMENTS. In cases where there are doubts, I remind you of a useful trick: we take the experimental value "x", for example, and try (mentally or on a draft) to substitute this value into the "terrible expression".
1) First we need to calculate the expression, so the sum is the deepest nesting.
2) Then you need to calculate the logarithm:
4) Then cube the cosine:
5) At the fifth step, the difference:
6) And finally, the outermost function is the square root: 
Complex Function Differentiation Formula 
are applied in reverse order, from the outermost function to the innermost. We decide:
Seems to be no error...
(1) We take the derivative of the square root.
(2) We take the derivative of the difference using the rule 
(3) The derivative of the triple is equal to zero. In the second term, we take the derivative of the degree (cube).
(4) We take the derivative of the cosine.
(5) We take the derivative of the logarithm.
(6) Finally, we take the derivative of the deepest nesting .
It may seem too difficult, but this is not the most brutal example. Take, for example, Kuznetsov's collection and you will appreciate all the charm and simplicity of the analyzed derivative. I noticed that they like to give a similar thing at the exam to check whether the student understands how to find the derivative of a complex function, or does not understand.
The following example is for a standalone solution.
Example 3
Find the derivative of a function
Hint: First we apply the rules of linearity and the rule of differentiation of the product
Full solution and answer at the end of the lesson.
It's time to move on to something more compact and prettier.
It is not uncommon for a situation where the product of not two, but three functions is given in an example. How to find the derivative of the product of three factors?
Example 4
Find the derivative of a function 
First, we look, but is it possible to turn the product of three functions into a product of two functions? For example, if we had two polynomials in the product, then we could open the brackets. But in this example, all functions are different: degree, exponent and logarithm.
In such cases, it is necessary successively apply the product differentiation rule 
twice
The trick is that for "y" we denote the product of two functions: , and for "ve" - the logarithm:. Why can this be done? Is it 
- this is not the product of two factors and the rule does not work?! There is nothing complicated:
Now it remains to apply the rule a second time 
to bracket:
You can still pervert and take something out of the brackets, but in this case it is better to leave the answer in this form - it will be easier to check.
The above example can be solved in the second way: 
Both solutions are absolutely equivalent.
Example 5
Find the derivative of a function
This is an example for an independent solution, in the sample it is solved in the first way.
Consider similar examples with fractions.
Example 6
Find the derivative of a function 
Here you can go in several ways:
Or like this:
But the solution can be written more compactly if, first of all, we use the rule of differentiation of the quotient 
, taking for the whole numerator:
In principle, the example is solved, and if it is left in this form, it will not be a mistake. But if you have time, it is always advisable to check on a draft, but is it possible to simplify the answer? We bring the expression of the numerator to a common denominator and get rid of the three-story fraction:
The disadvantage of additional simplifications is that there is a risk of making a mistake not when finding a derivative, but when banal school transformations. On the other hand, teachers often reject the task and ask to “bring it to mind” the derivative.
A simpler example for a do-it-yourself solution:
Example 7
Find the derivative of a function
We continue to master the techniques for finding the derivative, and now we will consider a typical case when a “terrible” logarithm is proposed for differentiation
Example 8
Find the derivative of a function
Here you can go a long way, using the rule of differentiation of a complex function:
But the very first step immediately plunges you into despondency - you have to take an unpleasant derivative of a fractional degree, and then also from a fraction.
That's why before how to take the derivative of the “fancy” logarithm, it is previously simplified using well-known school properties:
! If you have a practice notebook handy, copy these formulas right there. If you don't have a notebook, draw them on a piece of paper, as the rest of the lesson's examples will revolve around these formulas.
The solution itself can be formulated like this:
Let's transform the function:
We find the derivative: 
The preliminary transformation of the function itself greatly simplified the solution. Thus, when a similar logarithm is proposed for differentiation, it is always advisable to “break it down”.
And now a couple of simple examples for an independent solution:
Example 9
Find the derivative of a function 
Example 10
Find the derivative of a function
All transformations and answers at the end of the lesson.
logarithmic derivative
If the derivative of logarithms is such sweet music, then the question arises, is it possible in some cases to organize the logarithm artificially? Can! And even necessary.
Example 11
Find the derivative of a function 
Similar examples we have recently considered. What to do? One can successively apply the rule of differentiation of the quotient, and then the rule of differentiation of the product. The disadvantage of this method is that you get a huge three-story fraction, which you don’t want to deal with at all.
But in theory and practice there is such a wonderful thing as the logarithmic derivative. Logarithms can be organized artificially by "hanging" them on both sides:
Note
: because function can take negative values, then, generally speaking, you need to use modules: 
, which disappear as a result of differentiation. However, the current design is also acceptable, where by default the complex values. But if with all rigor, then in both cases it is necessary to make a reservation that.
Now you need to “break down” the logarithm of the right side as much as possible (formulas in front of your eyes?). I will describe this process in great detail:
Let's start with the differentiation.
We conclude both parts with a stroke:
The derivative of the right side is quite simple, I will not comment on it, because if you are reading this text, you should be able to handle it with confidence.
What about the left side?
On the left side we have complex function. I foresee the question: “Why, is there one letter “y” under the logarithm?”.
The fact is that this "one letter y" - IS A FUNCTION IN ITSELF(if it is not very clear, refer to the article Derivative of a function implicitly specified). Therefore, the logarithm is an external function, and "y" is an internal function. And we use the compound function differentiation rule 
:
On the left side, as if by magic, we have a derivative. Further, according to the rule of proportion, we throw the “y” from the denominator of the left side to the top of the right side:
And now we remember what kind of "game"-function we talked about when differentiating? Let's look at the condition: 
Final answer: 
Example 12
Find the derivative of a function
This is a do-it-yourself example. Sample design of an example of this type at the end of the lesson.
With the help of the logarithmic derivative, it was possible to solve any of examples No. 4-7, another thing is that the functions there are simpler, and, perhaps, the use of the logarithmic derivative is not very justified.
Derivative of exponential function
We have not considered this function yet. An exponential function is a function that has and the degree and base depend on "x". A classic example that will be given to you in any textbook or at any lecture:
How to find the derivative of an exponential function?
It is necessary to use the technique just considered - the logarithmic derivative. We hang logarithms on both sides:
As a rule, the degree is taken out from under the logarithm on the right side:
As a result, on the right side we have a product of two functions, which will be differentiated according to the standard formula 
.
We find the derivative, for this we enclose both parts under strokes:
The next steps are easy:
Finally: 
If some transformation is not entirely clear, please re-read the explanations of Example 11 carefully.
In practical tasks, the exponential function will always be more complicated than the considered lecture example.
Example 13
Find the derivative of a function
We use the logarithmic derivative. 
On the right side we have a constant and the product of two factors - "x" and "logarithm of the logarithm of x" (another logarithm is nested under the logarithm). When differentiating a constant, as we remember, it is better to immediately take it out of the sign of the derivative so that it does not get in the way; and, of course, apply the familiar rule 
:
The operation of finding a derivative is called differentiation.
As a result of solving problems of finding derivatives of the simplest (and not very simple) functions by defining the derivative as the limit of the ratio of the increment to the increment of the argument, a table of derivatives and precisely defined rules of differentiation appeared. Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716) were the first to work in the field of finding derivatives.
Therefore, in our time, in order to find the derivative of any function, it is not necessary to calculate the above-mentioned limit of the ratio of the increment of the function to the increment of the argument, but only need to use the table of derivatives and the rules of differentiation. The following algorithm is suitable for finding the derivative.
To find the derivative, you need an expression under the stroke sign break down simple functions and determine what actions (product, sum, quotient) these functions are related. Further, we find the derivatives of elementary functions in the table of derivatives, and the formulas for the derivatives of the product, sum and quotient - in the rules of differentiation. The table of derivatives and differentiation rules are given after the first two examples.
Example 1 Find the derivative of a function
Solution. From the rules of differentiation we find out that the derivative of the sum of functions is the sum of derivatives of functions, i.e.
From the table of derivatives, we find out that the derivative of "X" is equal to one, and the derivative of the sine is cosine. We substitute these values in the sum of derivatives and find the derivative required by the condition of the problem:
Example 2 Find the derivative of a function
Solution. Differentiate as a derivative of the sum, in which the second term with a constant factor, it can be taken out of the sign of the derivative:
If there are still questions about where something comes from, they, as a rule, become clear after reading the table of derivatives and the simplest rules of differentiation. We are going to them right now.
Table of derivatives of simple functions
| 1. Derivative of a constant (number). Any number (1, 2, 5, 200...) that is in the function expression. Always zero. This is very important to remember, as it is required very often | |
| 2. Derivative of the independent variable. Most often "x". Always equal to one. This is also important to remember | |
| 3. Derivative of degree. When solving problems, you need to convert non-square roots to a power. | |
| 4. Derivative of a variable to the power of -1 | |
| 5. Derivative of the square root | |
| 6. Sine derivative | |
| 7. Cosine derivative |  |
| 8. Tangent derivative |  |
| 9. Derivative of cotangent |  |
| 10. Derivative of the arcsine |  |
| 11. Derivative of arc cosine |  |
| 12. Derivative of arc tangent |  |
| 13. Derivative of the inverse tangent |  |
| 14. Derivative of natural logarithm | |
| 15. Derivative of a logarithmic function |  |
| 16. Derivative of the exponent | |
| 17. Derivative of exponential function |
Differentiation rules
| 1. Derivative of the sum or difference |  |
| 2. Derivative of a product |  |
| 2a. Derivative of an expression multiplied by a constant factor | |
| 3. Derivative of the quotient |  |
| 4. Derivative of a complex function |  |
Rule 1If functions
are differentiable at some point , then at the same point the functions
and
those. the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions.
Consequence. If two differentiable functions differ by a constant, then their derivatives are, i.e.
Rule 2If functions
are differentiable at some point , then their product is also differentiable at the same point
and
those. the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other.
Consequence 1. The constant factor can be taken out of the sign of the derivative:
Consequence 2. The derivative of the product of several differentiable functions is equal to the sum of the products of the derivative of each of the factors and all the others.
For example, for three multipliers:
Rule 3If functions
differentiable at some point And , then at this point their quotient is also differentiable.u/v , and
those. the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator.
Where to look on other pages
When finding the derivative of the product and the quotient in real problems, it is always necessary to apply several differentiation rules at once, so more examples on these derivatives are in the article."The derivative of a product and a quotient".
Comment. You should not confuse a constant (that is, a number) as a term in the sum and as a constant factor! In the case of a term, its derivative is equal to zero, and in the case of a constant factor, it is taken out of the sign of the derivatives. This is a typical mistake that occurs at the initial stage of studying derivatives, but as the average student solves several one-two-component examples, the average student no longer makes this mistake.
And if, when differentiating a product or a quotient, you have a term u"v, in which u- a number, for example, 2 or 5, that is, a constant, then the derivative of this number will be equal to zero and, therefore, the entire term will be equal to zero (such a case is analyzed in example 10).
Another common mistake is the mechanical solution of the derivative of a complex function as the derivative of a simple function. That's why derivative of a complex function devoted to a separate article. But first we will learn to find derivatives of simple functions.
Along the way, you can not do without transformations of expressions. To do this, you may need to open in new windows manuals Actions with powers and roots And Actions with fractions .
If you are looking for solutions to derivatives with powers and roots, that is, when the function looks like 
, then follow the lesson " Derivative of the sum of fractions with powers and roots".
If you have a task like 
, then you are in the lesson "Derivatives of simple trigonometric functions".
Step by step examples - how to find the derivative
Example 3 Find the derivative of a function
Solution. We determine the parts of the function expression: the entire expression represents the product, and its factors are sums, in the second of which one of the terms contains a constant factor. We apply the product differentiation rule: the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other:
Next, we apply the rule of differentiation of the sum: the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions. In our case, in each sum, the second term with a minus sign. In each sum, we see both an independent variable, the derivative of which is equal to one, and a constant (number), the derivative of which is equal to zero. So, "x" turns into one, and minus 5 - into zero. In the second expression, "x" is multiplied by 2, so we multiply two by the same unit as the derivative of "x". We get the following values of derivatives:
We substitute the found derivatives into the sum of products and obtain the derivative of the entire function required by the condition of the problem:
And you can check the solution of the problem on the derivative on .
Example 4 Find the derivative of a function
Solution. We are required to find the derivative of the quotient. We apply the formula for differentiating a quotient: the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator. We get:
We have already found the derivative of the factors in the numerator in Example 2. Let's also not forget that the product, which is the second factor in the numerator in the current example, is taken with a minus sign:
If you are looking for solutions to such problems in which you need to find the derivative of a function, where there is a continuous pile of roots and degrees, such as, for example, 
then welcome to class "The derivative of the sum of fractions with powers and roots" .
If you need to learn more about the derivatives of sines, cosines, tangents and other trigonometric functions, that is, when the function looks like , then you have a lesson "Derivatives of simple trigonometric functions" .
Example 5 Find the derivative of a function
Solution. In this function, we see a product, one of the factors of which is the square root of the independent variable, with the derivative of which we familiarized ourselves in the table of derivatives. According to the product differentiation rule and the tabular value of the derivative of the square root, we get:
You can check the solution of the derivative problem on derivative calculator online .
Example 6 Find the derivative of a function
Solution. In this function, we see the quotient, the dividend of which is the square root of the independent variable. According to the rule of differentiation of the quotient, which we repeated and applied in example 4, and the tabular value of the derivative of the square root, we get:
To get rid of the fraction in the numerator, multiply the numerator and denominator by .
